Solving Multi-Step Inequalities

- What You ll Learn To solve multi-step inequalities with variables on one side To solve multi-step inequalities with variables on both sides... And Why To find the measurements of a banner, as in Eample Solving Multi-Step Inequalities Check Skills You ll Need for Help Lessons - and - Solve each equation, if possible. If the equation is an identity or if it has no solution, write identity or no solution.. (c ) 6. t 6 (t ) no solution. p 9 p. 7n n (n ) identity. k k 7 6 9 6. t t Find the missing dimension of each rectangle. 7. perimeter = 0 cm 0 cm 8. perimeter = 78 in. in. cm 6 in. w -. Plan Objectives To solve multi-step inequalities with variables on one side To solve multi-step inequalities with variables on both sides Eamples Using More Than One Step Real-World Problem Solving Using the Distributive Property Gathering Variables on One Side of an Inequality Multi-Step Inequalities Part Solving Variables Inequalities on One Side With Variables on One Side Sometimes you need to perform two or more steps to solve an inequality. Models can help you understand how to solve multi-step inequalities. Math Background Many real world eamples are modeled by these more comple inequalities. Students should have several different ways of solving many eercises., The tiles model the inequality. More Math Background: p. 98C, Add to each side. Lesson Planning and Resources See p. 98E for a list of the resources that support this lesson.,,, Simplify by removing the zero pairs. Divide each side into two equal groups. Each green tile is less than two yellow tiles, so R. Lesson - Solving Multi-Step Inequalities 9 Bell Ringer Practice Check Skills You ll Need For intervention, direct students to: Solving Two-Step Equations Lesson -: Eample Etra Skills and Word Problems Practice, Ch. Solving Multi-Step Equations Lesson -: Eamples, Etra Skills and Word Problems Practice, Ch. Special Needs L Have students solve multi-step inequality problems in the same way as they would multi-step equations. Point out that they will only change the inequality symbol if they multiply or divide by a negative number. Below Level L As the inequalities become more comple, reassure students that, just like multi-step equations, multi-step inequalities can be solved by performing one step at a time. 9

. Teach To solve inequalities, you undo addition and subtraction first. Then undo multiplication and division. Guided Instruction Using More Than One Step Math Tip Point out to students that they are performing the order of operations in reverse to isolate the variable on one side of the equation. Teaching Tip Ask students to list some reasonable widths for the banner and some unreasonable widths, such as 6 inches, that meet the criteria of the solution. Visual Learners Some students may forget to distribute the factor (outside the parentheses) to the inside the parentheses, and some students may distribute it to the t. Have students draw arrows from the factor to both terms inside the parentheses. Additional Eamples Solve + b. Check the solution. b R The band is making a rectangular banner that is 0 feet long with trim around the edges. What are the possible widths the banner can be if there is no more than 8 feet of trim? feet or less Solve + (6 - ). S 0 Chapter Solving Inequalities Solve 7 6a. 9. Check the solution. 7 6a 7. 9 7 Subtract 7 from each side. 6a. 6a 6. 6 Divide each side by 6. a. Check 7 6a 9 Check the computation. 7 6() 0 9 Substitute for a. 9 9 7 6a. 9 Check the direction of the inequality. 7 6(). 9 Substitute for a.. 9 Check your solution. a. # L 6 b., 7 t t R c. 8, n n S You can adapt familiar formulas like the formula for the perimeter of a rectangle to write inequalities. You determine which inequality symbol to use from the realworld situation. Real-World Problem Solving Geometry The school band needs a banner to carry in a parade. The banner committee decides that the length of the banner should be 8 feet. A committee member drew the diagram at the left to help understand the problem. What are the possible widths of the banner if they can use no more than 8 feet of trim? Relate Since the border goes around the edges of a rectangular banner, you can adapt the perimeter formula P O w. twice the length plus Write (8) twice the width w (8) w # 8 6 w # 8 Simplify (8). 6 w 6 # 8 6 Subtract 6 from each side. w # w # Divide each side by. w # 6 The banner s width must be 6 feet or less. can be no more than To make a second banner, the committee decided to make the length feet. They have 0 feet of a second type of trim. Write and solve an inequality to find the possible widths of the second banner. () ± w K 0, so the banner s width must be 8 feet or less. # the length of trim 8 0 Advanced Learners L Have students write a multi-step inequality with variables on both sides and that requires the Distributive Property. Let them echange inequalities and solve. English Language Learners ELL Be sure that students understand the word problems, as in Eample. Have them eplain to a partner what they need to do, and give the equivalent math sentence.

Using the Distributive Property Solve (t ) t $. t t $ Use the Distributive Property. t $ Combine like terms. t $ t $ Subtract from each side. t # Divide each side by. Reverse the inequality symbol. t # Part Solving Variables Inequalities on Both Sides With Variables on Both Sides A B C D E A B C D E A B C D E A B C D E A B C D E B C D E Test-Taking Tip To grid a mied number, write it as an improper fraction:. 6 nline Visit: PHSchool.com Web Code: ate-077 6 / / /.... 0 0 0 6 6 6 6 7 7 7 7 8 8 8 8 9 9 9 9 Check your solution. a. p (p 7), 8 b. # (m 7) c. 8. ( b) p R m K b S Many inequalities have variables on both sides of the inequality symbol. You need to gather the variable terms on one side of the inequality and the constant terms on the other side. Solve 6z, z. 6z z, z z z, z, z, 6 Gathering Variables on One Side of an Inequality To gather variables on the left, subtract z from each side. Combine like terms. To gather the constants on the right, add to each side. z 6, Divide each side by. z, Solve b. 7 b. Check your solution. b S Multi-Step Inequalities Gridded Response Solve ( m) $ (m ). m $ 8m 8 Use the Distributive Property. m 8m $ 8m 8 8m Subtract 8m from each side. m $ 8 Combine like terms. m $8 Add to each side. m $6 m # 6 m # Divide each side by. Reverse the inequality symbol. Solve 6( ) $ 7( ). Check your solution. K Alternative Method Work the eample a second way on the board by subtracting 6z from both sides. Lead students to see that division by a negative number is unnecessary if you choose the operation that results in a positive coefficient. Then, you do not have to remember to reverse the inequality sign. Additional Eamples Solve 8z - 6 z +. z R Solve (- + d) # (d - ). d L Resources Daily Notetaking Guide - L Daily Notetaking Guide - Adapted Instruction L Closure Instruct students to write a multistep inequality with variables on both sides that requires the use of the distributive property. Have students echange inequalities and solve. Lesson - Solving Multi-Step Inequalities

. Practice Assignment Guide A B -, -0, 70-7 A B, -69, 7-7 C Challenge 7-79 Test Prep 80-8 Mied Review 8-98 Homework To check students understanding of key skills and concepts, go over Eercises 0, 8,,,. Error Prevention! Eercises 9 Simple mistakes can be made while checking a solution, thus making you think your answer is incorrect when it is correct. Remind students that it is easiest to check an answer by substituting 0 for the variable in each solution. GPS Enrichment Guided Problem Solving Reteaching Adapted Practice Practice Pearson Education, Inc. All rights reserved. Name Class Date Practice - Percent of Change Find each percent of change. Describe the percent of change as an increase or decrease. Round to the nearest whole number.. 6 g to 7 g. 0 cm to 00 cm. 90 in. to in.. 00 lb to 00 lb. $90 to $8.0 6. $00 to $0 7. $ to $.0 8. 00 mi to 7 mi 9. 80 m to 0 m 0. 8 to 76. 60 to 0. 600 mi to 80 mi. 8 to 7. 90 yd to 6 yd. 6. to.8 6. $8.0 to $.7 7. 6 to 9 8. 7 to 66 9. 6 to 8 0. to 8 Find each percent of change. Describe the percent of change as an increase or decrease. Round to the nearest whole number.. In 98, the average price for gasoline was $.0/gal. In 000, the average price for gasoline was $.6. Find the percent of change.. In 980, Teas had 7 U.S. Representatives. That number increased to 0 in 000. Find the percent of change.. In 980, the average annual tuition charge for a four-year public university was $80. The average annual tuition charge in 000 was $6. What is the percent of change?. The United States imported 6,909,000 barrels of oil per day in 980. In 000, the United States imported,9,000 barrels of oil per day.what is the percent of change?. In 977, the average number of households with cable television was 6.6%. In 000, the average number of households with cable television was 68%. What is the percent of change? 6. In 989, there were 8,000 licensed drivers under the age of 6. In 999, the total number of licensed drivers under 6 was,8. Find the percent of change. 7. In 990,Atlanta, GA, failed to meet air quality standards on days. In 999, Atlanta failed to meet air quality standards on 6 days. What is the percent of change? Find the greatest possible error and the percent error for each measurement. 8. cm 9. 0. cm 0. 6 cm. 6 in.. 6.8 g. 0.9 cm Find the minimum and maimum possible areas for rectangles with the following measurements.. 8 cm 0 cm. in. in. 6. 8 m m Find the minimum and maimum possible volume for a rectangular solid with the following measurements. L L L L L EXERCISES EXERCISES practice, see Etra Practice. Practice and Problem Solving A B Practice by Eample for Help Eample (page 0) Eample (page 0) Eample (page ) Eample (page ) Eample (page ) Apply Your Skills 9. Add and subtract y from each side. 0. Add to each side, then multiply each side by and reverse the inequality sign. Chapter Solving Inequalities For more For more eercises, see Etra Skill and Word Problem Practice. Check your solution.. d 7 # d K. m.8 m S., 8 S. n $ n K. 8 # q q L 6. # h h L 7. 7 # a a K 8. 0. 9 b b S 6 9. 9c. c R Write and solve an inequality. 0. On a trip from Virginia to Florida, the Sampson family wants to travel at least 0 miles in 8 hours of driving. What must be their average rate of speed? 8t L 0 and t L., so the average rate of speed must be at least. mi/h.. Geometry The perimeter of an isosceles triangle is at most 7 cm. One side is 8 cm long. Find the possible lengths of the two congruent sides. 7 L s ± 8 and s K 9., so the two equal sides must be no longer than 9. cm.. You want to solve an inequality containing the epression ( ). The net line in your solution would rewrite this epression as 9. 6 ± 9. (j ) $6 j L. (6b ). 0 b R. (h ), h S 6. # ( 6) L 7..(y 7) y S 88. (w ) # 0 w K 9. (c ). 7 0. (r ) 7 $ 8. 6 # (n ) c R 7 r K n K 9. w, w w R. t 7 $ t 9 t K. d 7 $ d d K 6. n # n n L 6. k # k 9 k L 7. s 6. 6 s s R 0 8. 6p. p 8 p S 9.. 6 0. m, m S m S. (v ) $ v. q 6 # (q ). ( r) $ r v L q K r L. 9, 7 ( ). (m 8),8 m 6. v # (v 6) R m S 8 v L Tell what you must do to the first inequality in order to get the second. Subtract 8 Subtract 7 from 7. 8 s. 6; s. 8 8. g 7 $ 9; from each side. g $ each side. z 9. y. 9 y; y. See left. 0. 8. ; 0, z See left.. Writing Suppose a friend is having difficulty solving.( p ). ( p ). Eplain how to solve the inequality, showing all necessary steps and identifying the properties you would use. See margin.. Multiple Choice Mandela is starting a part-time wordprocessing business out of his home. He plans to charge $ per hour. The table at the right shows his epected monthly business epenses. Which inequality describes the number of hours h he must work in a month to make a profit of at least $600? C h $ 600 h 600 $ 90 6 h $ 90 6 600 h 600 # 90 6 Epense Cost Equipment rental Materials Business phone $90 $ $6 7. 6 in. in. 8 in. 8. cm cm 8 cm 9. m m m pages 9 Eercises. Answers may vary. Sample: To solve.(p ) S (p ± ), first use the Distributive Property to simplify both sides. The result is.p 0 S p ± 6. Then use the Subtraction Property of Inequality. Subtract 6 from each side and.p from each side. The result is 6 S 0.p. Then use the Division Property of Inequality. Divide each side by 0.. The result is S p. So the solution is S p or p R.

Real-World Connection Normal blood pressure for teens is about 0/70.. For the number of guests, (0.7)00 ±. L 0, L 80, so at least 80 guests must attend. Vocabulary Tip Inequalities or equations that are always true are called identities. (See p. 6). Epenses The sophomore class is planning a picnic. The cost of a permit to use GPS a city park is $0. To pay for the permit, there is a fee of $.7 for each sophomore and $. for each guest who is not a sophomore. Two hundred sophomores plan to attend. Write and solve an inequality to find how many guests must attend for the sophomores to pay for the permit. See left.. Health Care Systolic blood pressure is the higher number in a blood pressure reading. It is measured as your heart muscle contracts. The formula P # a 0 gives the normal systolic blood pressure P based on age a. a. At age 0, does 0 represent a maimum or a minimum normal systolic pressure? maimum b. Find the normal systolic blood pressure for a 0-year-old person. no more than Match each inequality with its graph below... B 6.. E 7.. F 8.. A 9.. D 0. ( ). C A. B. 0 0 C. D. 0 0 E. F. 0 0. Open-Ended Write two different inequalities that you can solve by adding and multiplying by. Answers may vary. Samples: S 0, R ; K 0, L. r, r r r R. m # m m L. (0. s) $(.s) s K.. n $ 8 6 n n K 8 6. (8 s), 0 s R 8 7..8 k #. k k K. 8. 0. (n ) (n ) n S 9. (r ) (r ) # r K 60. ( 7). (7 ) S 6. (a ) 6a #9 a L 6. (m ) $ (m ) m L 6. 7 (k ) $ (k ) k K 6 6. n (n ) # n L 6. ( 8) # K 0 66. a (a ), 0 a R 6 67. c (c ) $ ( c) c L 68. a. Solve t # 8t by gathering the variable terms on the left side and the constant terms on the right side of the inequality. t K 9, t L b. Solve t # 8t by gathering the constant terms on the left side and the variable terms on the right side of the inequality. 9 K t, t L c. Compare the results of parts (a) and (b). The results are the same. Connection to Geometry Eercise Some students may want to use the formula for the perimeter of a rectangle. Tell them there is no set formula for the perimeter of a triangle. Ask: What is special about an isosceles triangle? Two sides are equal. Instruct students to keep this in mind when they are writing their inequality. Eercise Normal systolic blood pressure for an adult is from 90 to 0. nline Homework Help Visit: PHSchool.com Web Code: ate-00 69. a. Mental Math Like equations, some inequalities are true for all values of the variable, and some inequalities are not true for any values of the variable. Determine whether each inequality is always true or never true. See margin. i. s 6 $ 6 s ii. r. r iii. (n ), n b. Critical Thinking How can you tell whether an inequality is always true or never true without solving? See margin. Lesson - Solving Multi-Step Inequalities 69a. i. always true ii. always true iii. never true b. If the coefficients of the only variable on each side of an inequality are the same, then the inequality will either be always true or never true.

. Assess & Reteach Lesson Quiz. 8 + a $ a L. - p p - 6 p S 7. ( - ) + 7 R 9. (c + ) # (c - ) c K Alternative Assessment Have each student graph - on a number line. Instruct each student to work backwards from this inequality to write an inequality that can be solved using at least three steps. 70. For amount of sales, 0 ± 0.0 L 60, so to reach her goal, she must have at least $7000 in sales. 7a. R n; the greater values in the table make the inequality false. b. Sample:. c. R ; this is close to the estimate. Real-World for Help For a guide to solving Eercise 7, see p. 6. C.6 Challenge Connection The Parthenon, an ancient Greek temple, has dimensions that form a golden rectangle. 70. Commission Joleen is a sales associate in a clothing store. Each week she earns $0 plus a commission equal to % of her sales. This week her goal is to earn no less than $60. Write and solve an inequality to find the dollar amount of the sales she must have to reach her goal. See left. 7. A student uses the table below to help solve 6, ( ). 0 6 6(0) 0. 6(0.) 6() 7. 6(.) 0 true true true false ( ) ( 0) ( 0.). ( ) 0 (.) 7. a. Critical Thinking Based on the table, would you epect the solution of 6, ( ) to be of the form, n or. n? Eplain. b. Estimate Based on the table, estimate the value of n. c. Solve the inequality. Compare the actual solution to your estimated solution. Error Analysis Find and correct the mistake in each student s work. 7. 7. (n ) n n n n n n See back of book. c b 7. a. Solve a b. c for, where a is positive. S a c b b. Reasoning Solve a b. c for, where a is negative. R a 7. Geometry The base of a triangle is 0 in. Its height is ( ) in. Its area is no more than 6 in.. What are the possible integer values of? is an integer between and 7, inclusive. 76. Architecture The rectangle shown on the building at the left is a golden rectangle. Artists often use the golden rectangle because they consider it to be pleasing to the eye. The ratio of two sides of a golden rectangle is approimately :.6. Suppose you are making a picture frame in the shape of a golden rectangle. You have a 6-in. length of wood to use for a frame. What are the dimensions of the largest frame you can make? Round to the nearest tenth of an inch. 8.8 in.. in. 77. Critical Thinking Find a value of a such that the number line below shows all the solutions of a #. 0 6 7 8 9 78. Earning You can earn money by handing out flyers in the afternoon for $6.0 an hour and by typing a newsletter in the evening for $8 an hour. You have 0 hours available to work. What are the greatest number of hours you can spend handing out flyers and still make at least $? 0 h 79. Freight Handling The freight elevator of a building can safely carry a load of at most 000 lb. A worker needs to move supplies in 0-lb boes from the loading dock to the fourth floor of the building. The worker weighs 60 lb. The cart she uses weighs 9 lb. a. What is the greatest number of boes she can move in one trip? 7 boes b. The worker must deliver 0 boes to the fourth floor. How many trips must she make? trips Chapter Solving Inequalities

Standardized Test Prep Test Prep Short Response Mied Mied Review Review for Help Multiple Choice Lesson - 80. The Science Club hopes to collect at least 00 kg of aluminum cans for recycling this semester ( weeks). The graph at the right shows the first week s results. Let represent the average mass of cans required per week for the remainder of the semester. Which inequality would you use to find? C 00 (00 8) A. $ B. $ Mon. Wed. Day (008) C. $ 0 D.. Q 00 0 R 8 8. Solve 8.. F F., G.. H., J.. 8. Solve n 6 # 7n. A A. n # 8 B. n $ 8 C. n # 8 D. n $ 8 8. Great Gifts pays its supplier $6 for each bo of bells. The owner wants to determine the least amount he can charge his customers per bell in order to make at least a 0% profit per bo. Which inequality should he use? F F. $.0(6) G. 6 #.0() H. 0.0() $ 6 J. 0.0() # 6 8. Mawell orders at least 0 bottles of flea shampoo per month for his petgrooming business. His supplier charges $ per quart bottle plus a $ handling fee per order. A competing supplier offers a similar product for $ per quart bottle plus a $ handling fee per order. The salesman for the competitor shows Mawell that 0 bottles from his company would cost only $ compared to $ from Mawell s current supplier. Which supplier would you advise Mawell to use? Eplain or show work to support your advice to Mawell. See margin. 8. 9m $ 6 m K 86. # yy L 8 87.. S 88. t # t L 89. b, 8 b R 7 90.. 7w w S 98 9. 6, p p S 9. 0.d $. d L 7 9.. 0 S 0 Mass (kg) 0 Aluminum Cans Collected in Week Fri. Test Prep Resources For additional practice with a variety of test item formats: Standardized Test Prep, p. 7 Test-Taking Strategies, p. Test-Taking Strategies with Transparencies Eercise 80 Remind students to look for key words and phrases, such as remainder of the semester. pages Eercises 8. [] Mawell should continue ordering from the current supplier. Let number of bottles Mawell orders per month. The cost from the current supplier is ±. The cost from the competitor is ±. The inequality ± K ± represents the number of bottles of shampoo for which the competitor will be less epensive. Since 0, the competitor will not be less epensive for orders of 0 bottles or more. (OR equivalent eplanation) [] incorrect answer OR insufficient eplanation Lesson -6 Lesson - 9. Your family leaves your town traveling at an average rate of mi/h. Two hours later, your neighbor leaves your town along the same road at an average rate of 60 mi/h. How many hours will it take your neighbor to overtake you? 6 h Simplify each epression. 9. 6 96. () 6 97. () () 98. 6 lesson quiz, PHSchool.com, Web Code: ata-00 Lesson - Solving Multi-Step Inequalities